Precise definition of linear combination

[redjoker]

i know that a linear combination of the vectors {v1,v2,...,vn} is any sum with terms that are scalar multiples of those vectors. But is a1v1 + a2v2 the same linear combination as a2v2 + a1v1? i know they evaluate to the same thing because vector addition is commutative but if i wanted to be precise would i say that it's the same linear combination or a different one only with the same set of coefficients? because if at least one of b1 and b2 was different from a1 and a2, then b1v1 + b2v2 is not considered to be the same linear combination even though they might be equal.

 

[tiny-tim]

Hi redjoker!

i know that a linear combination of the vectors {v1,v2,...,vn} is any sum with terms that are scalar multiples of those vectors. But is a1v1 + a2v2 the same linear combination as a2v2 + a1v1?

Yes!

And stop worrying … there really isn't a problem!

 

[mma]

The point is that linear combination refers to a relation between a vector, say x and a set of vectors say {x_i}
A good definition is of Paul Halmos (in Finite-dimensional vector spaces):

We shall say, whenever x = Σ_iα_ix_i, that x is a linear combination of {x_i}

In other words, the phrase "x is the linear combination of..." is the synonym of "x is linearly dependent on...".